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Given a hypergraph $\mathcal{H}$, the dual hypergraph of $\mathcal{H}$ is the hypergraph of all minimal transversals of $\mathcal{H}$. The dual hypergraph is always Sperner, that is, no hyperedge contains another. A special case of Sperner hypergraphs are the conformal Sperner hypergraphs, which correspond to the families of maximal cliques of graphs. All these notions play an important role in many fields of mathematics and computer science, including combinatorics, algebra, database theory, etc. In this paper we study conformality of dual hypergraphs. While we do not settle the computational complexity status of recognizing this property, we show that the problem is in co-NP and can be solved in polynomial time for hypergraphs of bounded dimension. In the special case of dimension $3$, we reduce the problem to $2$-Satisfiability. Our approach has an implication in algorithmic graph theory: we obtain a polynomial-time algorithm for recognizing graphs in which all minimal transversals of maximal cliques have size at most $k$, for any fixed $k$.

In this work, we propose an absolute value block $\alpha$-circulant preconditioner for the minimal residual (MINRES) method to solve an all-at-once system arising from the discretization of wave equations. Since the original block $\alpha$-circulant preconditioner shown successful by many recently is non-Hermitian in general, it cannot be directly used as a preconditioner for MINRES. Motivated by the absolute value block circulant preconditioner proposed in [E. McDonald, J. Pestana, and A. Wathen. SIAM J. Sci. Comput., 40(2):A1012-A1033, 2018], we propose an absolute value version of the block $\alpha$-circulant preconditioner. Our proposed preconditioner is the first Hermitian positive definite variant of the block $\alpha$-circulant preconditioner, which fills the gap between block $\alpha$-circulant preconditioning and the field of preconditioned MINRES solver. The matrix-vector multiplication of the preconditioner can be fast implemented via fast Fourier transforms. Theoretically, we show that for properly chosen $\alpha$ the MINRES solver with the proposed preconditioner has a linear convergence rate independent of the matrix size. To the best of our knowledge, this is the first attempt to generalize the original absolute value block circulant preconditioner in the aspects of both theory and performance. Numerical experiments are given to support the effectiveness of our preconditioner, showing that the expected optimal convergence can be achieved.

The complexity class Quantum Statistical Zero-Knowledge ($\mathsf{QSZK}$) captures computational difficulties of the time-bounded quantum state testing problem with respect to the trace distance, known as the Quantum State Distinguishability Problem (QSDP) introduced by Watrous (FOCS 2002). However, QSDP is in $\mathsf{QSZK}$ merely within the constant polarizing regime, similar to its classical counterpart shown by Sahai and Vadhan (JACM 2003) due to the polarization lemma (error reduction for SDP). Recently, Berman, Degwekar, Rothblum, and Vasudevan (TCC 2019) extended the $\mathsf{SZK}$ containment for SDP beyond the polarizing regime via the time-bounded distribution testing problems with respect to the triangular discrimination and the Jensen-Shannon divergence. Our work introduces proper quantum analogs for these problems by defining quantum counterparts for triangular discrimination. We investigate whether the quantum analogs behave similarly to their classical counterparts and examine the limitations of existing approaches to polarization regarding quantum distances. These new $\mathsf{QSZK}$-complete problems improve $\mathsf{QSZK}$ containments for QSDP beyond the polarizing regime and establish a simple $\mathsf{QSZK}$-hardness for the quantum entropy difference problem (QEDP) defined by Ben-Aroya, Schwartz, and Ta-Shma (ToC 2010). Furthermore, we prove that QSDP with some exponentially small errors is in $\mathsf{PP}$, while the same problem without error is in $\mathsf{NQP}$.

In the present paper, we study a multipoint boundary value problem for a system of Fredholm integro-differenial equations by the method of parameterization. The case of a degenerate kernel is studied separately, for which we obtain well-posedness conditions and propose some algorithms to find approximate and numerical solutions to the problem. Then we establish necessary and sufficient conditions for the well-posedness of the multipoint problem for the system of Fredholm integro-differential equations and develop some algorithms for finding its approximate solutions. These algorithms are based on the solutions of an approximating problem for the system of integro-differential equations with degenerate kernel.

We consider multi-variate signals spanned by the integer shifts of a set of generating functions with distinct frequency profiles and the problem of reconstructing them from samples taken on a random periodic set. We show that such a sampling strategy succeeds with high probability provided that the density of the sampling pattern exceeds the number of frequency profiles by a logarithmic factor. The signal model includes bandlimited functions with multi-band spectra. While in this well-studied setting delicate constructions provide sampling strategies that meet the information theoretic benchmark of Shannon and Landau, the sampling pattern that we consider provides, at the price of a logarithmic oversampling factor, a simple alternative that is accompanied by favorable a priori stability margins (snug frames). More generally, we also treat bandlimited functions with arbitrary compact spectra, and different measures of its complexity and approximation rates by integer tiles. At the technical level, we elaborate on recent work on relevant sampling, with the key difference that the reconstruction guarantees that we provide hold uniformly for all signals, rather than for a subset of well-concentrated ones. This is achieved by methods of concentration of measure formulated on the Zak domain.

We study the multivariate deconvolution problem of recovering the distribution of a signal from independent and identically distributed observations additively contaminated with random errors (noise) from a known distribution. For errors with independent coordinates having ordinary smooth densities, we derive an inversion inequality relating the $L^1$-Wasserstein distance between two distributions of the signal to the $L^1$-distance between the corresponding mixture densities of the observations. This smoothing inequality outperforms existing inversion inequalities. As an application of the inversion inequality to the Bayesian framework, we consider $1$-Wasserstein deconvolution with Laplace noise in dimension one using a Dirichlet process mixture of normal densities as a prior measure on the mixing distribution (or distribution of the signal). We construct an adaptive approximation of the sampling density by convolving the Laplace density with a well-chosen mixture of normal densities and show that the posterior measure concentrates around the sampling density at a nearly minimax rate, up to a log-factor, in the $L^1$-distance. The same posterior law is also shown to automatically adapt to the unknown Sobolev regularity of the mixing density, thus leading to a new Bayesian adaptive estimation procedure for mixing distributions with regular densities under the $L^1$-Wasserstein metric. We illustrate utility of the inversion inequality also in a frequentist setting by showing that an appropriate isotone approximation of the classical kernel deconvolution estimator attains the minimax rate of convergence for $1$-Wasserstein deconvolution in any dimension $d\geq 1$, when only a tail condition is required on the latent mixing density and we derive sharp lower bounds for these problems

I propose an alternative algorithm to compute the MMS voting rule. Instead of using linear programming, in this new algorithm the maximin support value of a committee is computed using a sequence of maximum flow problems.

In the area of query complexity of Boolean functions, the most widely studied cost measure of an algorithm is the worst-case number of queries made by it on an input. Motivated by the most natural cost measure studied in online algorithms, the competitive ratio, we consider a different cost measure for query algorithms for Boolean functions that captures the ratio of the cost of the algorithm and the cost of an optimal algorithm that knows the input in advance. The cost of an algorithm is its largest cost over all inputs. Grossman, Komargodski and Naor [ITCS'20] introduced this measure for Boolean functions, and dubbed it instance complexity. Grossman et al. showed, among other results, that monotone Boolean functions with instance complexity 1 are precisely those that depend on one or two variables. We complement the above-mentioned result of Grossman et al. by completely characterizing the instance complexity of symmetric Boolean functions. As a corollary we conclude that the only symmetric Boolean functions with instance complexity 1 are the Parity function and its complement. We also study the instance complexity of some graph properties like Connectivity and k-clique containment. In all the Boolean functions we study above, and those studied by Grossman et al., the instance complexity turns out to be the ratio of query complexity to minimum certificate complexity. It is a natural question to ask if this is the correct bound for all Boolean functions. We show a negative answer in a very strong sense, by analyzing the instance complexity of the Greater-Than and Odd-Max-Bit functions. We show that the above-mentioned ratio is linear in the input size for both of these functions, while we exhibit algorithms for which the instance complexity is a constant.

In this work we propose and analyze an extension of the approximate component mode synthesis (ACMS) method to the heterogeneous Helmholtz equation. The ACMS method has originally been introduced by Hetmaniuk and Lehoucq as a multiscale method to solve elliptic partial differential equations. The ACMS method uses a domain decomposition to separate the numerical approximation by splitting the variational problem into two independent parts: local Helmholtz problems and a global interface problem. While the former are naturally local and decoupled such that they can be easily solved in parallel, the latter requires the construction of suitable local basis functions relying on local eigenmodes and suitable extensions. We carry out a full error analysis of this approach focusing on the case where the domain decomposition is kept fixed, but the number of eigenfunctions is increased. The theoretical results in this work are supported by numerical experiments verifying algebraic convergence for the method. In certain, practically relevant cases, even exponential convergence for the local Helmholtz problems can be achieved without oversampling.

For an approximate solution of the non-stationary nonlinear Navier-Stokes equations for the flow of an incompressible viscous fluid, depending on the set of input data and the geometry of the domain, the area of optimal parameters in the variables $\nu$ and $\nu^{\ast}$ is experimentally determined depending on $\delta$ included in the definition of the $R_{\nu}$-generalized solution of the problem and the degree of the weight function in the basis of the finite element method. To discretize the problem in time, the Runge-Kutta methods of the first and second orders were used. The areas of optimal parameters for various values of the incoming angles are established.